Tensored Segre classes

We study a class obtained from the Segre class s(Z,Y) of an embedding of schemes by incorporating the datum of a line bundle on Z. This class satisfies basic properties analogous to the ordinary Segre class, but leads to remarkably simple formulas in standard intersection-theoretic situations such as excess or residual intersections. We prove a formula for the behavior of this class under linear joins, and use this formula to prove that a 'Segre zeta function' associated with ideals generated by forms of the same degree is a rational function.